MATH 5400, History of Mathematics

Transcription

1 MATH 5400, History of Mathematics Lecture 1: Introduction Professor: Peter Gibson September 8, 2016

2 The history of...what, exactly? P. Gibson (YorkU) Math / 16

3 The history of...what, exactly? What is mathematics? P. Gibson (YorkU) Math / 16

4 A proposed answer 1 Mathematics is the science of the abstract. P. Gibson (YorkU) Math / 16

5 A proposed answer 1 Mathematics is the science of the abstract. 2 It is a scholarly pursuit conducted with precision and rigour. P. Gibson (YorkU) Math / 16

6 A proposed answer 1 Mathematics is the science of the abstract. 2 It is a scholarly pursuit conducted with precision and rigour. 3 Mathematical ideas have been and continue to be central to cultural developments that have dramatically altered human society. P. Gibson (YorkU) Math / 16

7 The science of the abstract An abstract painting: P. Gibson (YorkU) Math / 16

8 The science of the abstract An abstract painting: Mark Rothko, Magenta, Black, Green on Orange, 1947 P. Gibson (YorkU) Math / 16

9 The science of the abstract An abstract painting: Mark Rothko, Magenta, Black, Green on Orange, 1947 Abstract refers to precisely defined abstract structures, such as numbers, arithmetic operations, lines, curves,... P. Gibson (YorkU) Math / 16

10 The science of the abstract An abstract painting: Mark Rothko, Magenta, Black, Green on Orange, 1947 Abstract refers to precisely defined abstract structures, such as numbers, arithmetic operations, lines, curves,...limits, differential equations... P. Gibson (YorkU) Math / 16

11 The science of the abstract An abstract painting: Mark Rothko, Magenta, Black, Green on Orange, 1947 Abstract refers to precisely defined abstract structures, such as numbers, arithmetic operations, lines, curves,...limits, differential equations..., groups, Riemannian manifolds, P. Gibson (YorkU) Math / 16

12 The science of the abstract An abstract painting: Mark Rothko, Magenta, Black, Green on Orange, 1947 Abstract refers to precisely defined abstract structures, such as numbers, arithmetic operations, lines, curves,...limits, differential equations..., groups, Riemannian manifolds,measure spaces, categories etc. P. Gibson (YorkU) Math / 16

13 The science of the abstract An abstract painting: Mark Rothko, Magenta, Black, Green on Orange, 1947 Abstract refers to precisely defined abstract structures, such as numbers, arithmetic operations, lines, curves,...limits, differential equations..., groups, Riemannian manifolds,measure spaces, categories etc., etc... P. Gibson (YorkU) Math / 16

14 Rigour and precision Precision and rigour confer a kind of timelessness that makes mathematics unique among scholarly pursuits. P. Gibson (YorkU) Math / 16

15 Rigour and precision Precision and rigour confer a kind of timelessness that makes mathematics unique among scholarly pursuits. An example of some old mathematics: al-karaji (c. 972) P. Gibson (YorkU) Math / 16

16 Rigour and precision Precision and rigour confer a kind of timelessness that makes mathematics unique among scholarly pursuits. An example of some old mathematics: al-karaji (c. 972) This is a area of current research! P. Gibson (YorkU) Math / 16

17 Some changes in human society... P. Gibson (YorkU) Math / 16

18 Some changes in human society... the first city states (civic administration) P. Gibson (YorkU) Math / 16

19 Some changes in human society... the first city states (civic administration) industrialization P. Gibson (YorkU) Math / 16

20 Some changes in human society... the first city states (civic administration) industrialization electricity (motors, artificial light, radio) P. Gibson (YorkU) Math / 16

21 Some changes in human society... the first city states (civic administration) industrialization electricity (motors, artificial light, radio) computers (so much for the slide rule...) P. Gibson (YorkU) Math / 16

22 Some changes in human society... the first city states (civic administration) industrialization electricity (motors, artificial light, radio) computers (so much for the slide rule...) the iphone (the information age in full swing) P. Gibson (YorkU) Math / 16

23 Some changes in human society... the first city states (civic administration) industrialization electricity (motors, artificial light, radio) computers (so much for the slide rule...) the iphone (the information age in full swing) These all have mathematical underpinnings. P. Gibson (YorkU) Math / 16

24 History History is bunk. Henry Ford How can we understand the history of mathematics? P. Gibson (YorkU) Math / 16

25 What is the purpose of history? P. Gibson (YorkU) Math / 16

26 What is the purpose of history? History tells us not only about the past, but also the present. P. Gibson (YorkU) Math / 16

27 An analogy: the study of genetics and evolution relates to both the past and present structure of organisms, simultaneously. P. Gibson (YorkU) Math / 16

28 An analogy: the study of genetics and evolution relates to both the past and present structure of organisms, simultaneously. (Ernst Haeckel) P. Gibson (YorkU) Math / 16

29 An analogy: the study of genetics and evolution relates to both the past and present structure of organisms, simultaneously. (Ernst Haeckel) Similarly, the history of human culture refers simultaneously to the past and present. P. Gibson (YorkU) Math / 16

30 An analogy: the study of genetics and evolution relates to both the past and present structure of organisms, simultaneously. (Ernst Haeckel) Similarly, the history of human culture refers simultaneously to the past and present. (Consider language: current usage embeds remnants of past culture.) P. Gibson (YorkU) Math / 16

31 The historical method We distinguish several types of sources: P. Gibson (YorkU) Math / 16

32 The historical method We distinguish several types of sources: primary sources secondary sources tertiary sources P. Gibson (YorkU) Math / 16

33 The historical method We distinguish several types of sources: primary sources secondary sources tertiary sources Three key questions: P. Gibson (YorkU) Math / 16

34 The historical method We distinguish several types of sources: primary sources secondary sources tertiary sources Three key questions: 1 What do we know? 2 How do we know it? 3 What don t we know? P. Gibson (YorkU) Math / 16

35 The historical method We distinguish several types of sources: primary sources secondary sources tertiary sources Three key questions: 1 What do we know? 2 How do we know it? 3 What don t we know? P. Gibson (YorkU) Math / 16

36 Some examples from the history of mathematics P. Gibson (YorkU) Math / 16

37 Some examples from the history of mathematics The dawn of civilization: P. Gibson (YorkU) Math / 16

38 Some examples from the history of mathematics The dawn of civilization: A clay tablet from Uruk (c. 31st century BC). P. Gibson (YorkU) Math / 16

39 A much more recent tablet: P. Gibson (YorkU) Math / 16

40 A much more recent tablet: Examples of Pythagorean triples (c. 18th century BC) in sexagesimal notation. P. Gibson (YorkU) Math / 16

41 Al-Karaji, again: P. Gibson (YorkU) Math / 16

42 Al-Karaji, again: On the construction of water wells (c. 10th century AD). P. Gibson (YorkU) Math / 16

43 , 8c/i t -= - A. tang. C }. ' wenn An eighteenth century example: Euler s translation of Benjamin Robins s work on ballistics. 364 BAS ZWEYTE CAPITEL VON DEM WIEDEKSTANDE DEE LUFT [ wovon das Integrale 1st Sfltut bet ARTILLERIE Setzt man nun Kurze halber = z, und e fur die Zahl, deren hyperbolischer Logarithmus == 1, so wird einer Unterfucfcuntj {i>d(^^te (am] famen bem ^nglif^en t>ea ^)rn. Benjamin 9vobin Die andere Aequation giebt dt~*~^ ton Seon^arb u(er r//_ ~ 4wc. t(, f ~~ 7 3 (& -h ) v yt? Man setze, um die Irrationalitat zu = heben, h aa, und t; = uu, so wird -4^c $aadii Snc/ du du\ 3 uu(aa-{-uu) 3 \aa + ww w^^/ wovon das Integrale zum Theil auf der Quadratur des Zirkuls beruht. Denn es ist /adu., w == A. tang., aa + 'uu D a der Eadius = 1 ge- das ist einem Zirkul-Bogen, dessen tangens =, nomruen wird. Also bekommt man ^\ 3 \a & a u J A, w.l Man setze nun wiederum a = Yh 9 und u = Yv, und bestimme die GrroBe C dergestalt, dab = t; b wird, wenn t == 0, so wird man finden oder (r- A. ^~ ~ A. -- A tang. 8 Yh yh tang. * Yh ; YV r ) yv \yjt P. Gibson (YorkU) Math / 16

44 Summary 1 Mathematics is the science of the abstract. 2 It is a scholarly pursuit conducted with precision and rigour. 3 Mathematical ideas have been and continue to be central to cultural developments that have dramatically altered human society. P. Gibson (YorkU) Math / 16

45 Summary 1 Mathematics is the science of the abstract. 2 It is a scholarly pursuit conducted with precision and rigour. 3 Mathematical ideas have been and continue to be central to cultural developments that have dramatically altered human society. the first city states (civic administration) industrialization electricity (motors, artificial light, radio) computers (so much for the slide rule...) the iphone (the information age in full swing) and much more... P. Gibson (YorkU) Math / 16

46 We distinguish several types of sources: primary sources secondary sources tertiary sources P. Gibson (YorkU) Math / 16

47 We distinguish several types of sources: primary sources secondary sources tertiary sources Three key questions: 1 What do we know? 2 How do we know it? 3 What don t we know? P. Gibson (YorkU) Math / 16